# Course grades. Computing Your Course Grade and GPA, or, Weighted Averages

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3 course. Take the following grades from a student's quarter: Course Units Grade Intro Statistics Psychology Biology I To nd the student's GPA for the quarter, we need to use the units as the weights (w i ) and the grade as the observation ( ): GP A x W i1 w i i1 w i We could also expand the table to aid in our calculations: 4.0(3.7) + 3.0(4.0) + 5.0(3.0) Course Units Grade U nits Grade Intro Statistics Psychology Biology I Total: GPA: 41.8 / Again, this is a dierent result than we would have obtained by taking the sample average: x The actual GPA is lower because the student received lower grades in the courses worth more units. We can also use weighed averages to compute cumulative GPAs without going down to the level of individual courses. Suppose that before starting the quarter above, the student had a cumulative GPA of 3.1 after taking a total of 48 units. We create the following table: Time Units GPA Cumulative Current quarter Again, we use the units as weights. Now, though, the GPAs are the observation: 48(3.1) + 12(3.48) cumulative GP A The student's cumulative GPA has gone up, but it is not right in the middle of the previous GPA and this quarter's GPA because this quarter's GPA is based o of one fourth the units of the previous cumulative calculation. 3

4 Investment portfolios Weighted averages aren't jused used in computing grades. Another use of weighted averages are in investment portfolios. When people or corporations invest in the stock or bond market, they invest dierent amounts of money in various securities. Many websites will show investments as a pie chart. Suppose you invested in ve companies, with amounts given below: Company Amount Invested AT&T \$5,000 Apple \$4,000 Google \$3,500 Starbucks \$3,500 Ford \$1,000 A pie chart of your portfolio would look like this: Apple AT&T Ford Google Starbucks Over the course of the year, you nd that AT&T went up 2%, Apple went up 10%, Google went down 4%, Starbucks went up 15%, and Ford went down 8%. How did your portfolio do overall? We want to nd the average change in our portfolio, so the individual stock changes are our observations, and the amount invested are the weights: Company Amount Invested (w i ) Change ( ) AT&T \$5,000 2.% Apple \$4, % Google \$3,500-4.% Starbucks \$3, % Ford \$1,000-8.% Now the total change for the portfolio is x W i1 w i i1 w i 5000(2) (10) ( 4) (15) ( 8) % 4

5 So you have gained 4.7% on your total investments. For comparison, taking the sample average of the changes gives 3.0. The weighted average is higher becuase the biggest loss came from the company in which you had the smallest investment, so it had a relatively small eect on your portfolio. Weighted average and sample mean The common sample mean x 1 n xi is actually a special case of a weighted average where all observations are given the same weight. Let's say we have n observations and we give each the weight w i 1. Then the weighted average is x W i1 w i i1 w i i1 1 i1 1 1 n i i1 n x i1 You can check for yourself that there is nothing special about setting w i 1. Any weight will work as long as its the same for all observations since the weights in the numerator will cancel with the weights in the denominator. Weighted averages can also be used in place of sample means to make problems quicker when the observations only have a few unique values. Suppose we take a sample of 10 adults and ask how many pets they have at home. The responses are: To compute the sample mean, we take x 1 n i1 Alternatively, we can create a frequency table: 0, 0, 1, 1, 1 1, 1, 2 2, 2 1 ( ) Number of pets Frequency Then compute the weighted average using the frequencies as weights: x W 2(0) + 5(1) + 3(2)

6 We get the same answer. This occurs because in the weighted average, we are combining all the observations with the same number together. Lets compute x dierently: x 1 n i1 1 ( ) 10 1 [2(0) + 5(1) + 3(2)] 10 We see this is exactly the same formula as the weighted average. Generally speaking, when observations can be collected into sets with the same value, we can use the frequency of each set as weights to compute a weighted average and avoid long summations. 6

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